1. Field of the Invention
The present invention relates to a nonlinear hysteresis control system for controlling an actuator, especially for a nanoscale actuator, such as made by piezoelectric ceramics, magnet, and quartz.
2. Description of the Related Art
The actuators made of, for example, piezoelectric ceramics, magnet, and quartz, etc. will present nonlinear hysteresis effect that may create difficulty for various control applications.
FIG. 1 shows a typical close-loop controller 90 which includes a computer 91, a D/A Converter 92, an amplifier 93 for outputting voltage and a measurement sensor 99. A user uses the computer 91 as an interface and applies a voltage on an actuator 94. The displacement of the actuator 94 will change due to different applied voltage, then the actuator 94 can move a working platform 95 (e.g. DNA inspection device). The displacement of the actuator 94/working platform 95 can be measured by the measurement sensor 99, so the user can adjust the output voltage to adjust the desired displacement of the actuator 94. However, the nanoscale measurement sensor 99 is extremely expensive, and therefore, referring to FIG. 2, a well-known open-loop controller 90a without the need measurement sensor 99 is developed and implemented as a feed forward controller. The control scheme is depicted in FIG. 3. A desired trajectory x(t) (i.e. a plurality of desired time corresponding to the plurality of desired displacement) is calculated by the well-known “inverse Preisach model algorithm codes 31a”, and appropriate applied voltage u(t) is computed to implement on the actuator 94 and the output displacement y(t) is then obtained.
To use “inverse Preisach model algorithm codes 31a”, the hysteresis effect profile 20a of the actuator 94 must be built (measured in advance) or recorded by the user is stored in the computer 91a. The hysteresis effect profile 20a can include various kinds of piezoelectric ceramics, magnet, and quartz, etc. in market, so that the user can use many type of actuators as long as corresponding profiles are stored in the computer 91a. 
The following paragraphs will briefly describe the classical Preisach Model and classical hysteresis effect profile hereafter so that the present invention can be understood well. Please also refer to “Mayergoyz, Mathematical Models of Hysteresis, Springer Verlag, New York, 1991.” for details
Preisach Model and Classical Hysteresis Effect Profile:
Hysteresis exists in the piezoelectric ceramics so that it increases the difficulty of controller design. To overcome this problem, a numerical Preisach model for describing the hysteresis of piezoelectric actuators is used. FIG. 4 shows an input-output diagram of the hysteresis operator, γαβ, defined with a value of 1 or 0. Parameters α and β correspond to the “up” and “down” switching values of the input voltage u(t). As u(t) increases to be greater than α, then γαβ is switched from 0 to 1. On the contrary, u(t) decreases to be smaller than β, then γαβ is switched from 1 to 0.
The Preisach model is expressed as follows:q(t)=∫∫α≧βμ(α,β)γαβ[u(t)]dαdβ  (1)where q(t) is the actuator displacement with respect to its natural length, and μ(α,β) is a weight function in the Preisach model. FIG. 5 illustrates how the Preisach works, where T0 represents the limiting triangle corresponding to the maximal displacement of the actuator. When the input voltage starts to increase from zero to some value u1, all hysteresis operators γαβ with switching values α less than u1 are switched to the “up” position in the region S+ as shown in FIG. 5A. As the input voltage starts to decrease from u1 to u2, all hysteresis operators γαβ with switching values β larger than u2 are switched to the “down” position, and the region S+ will be decreased as shown in FIG. 5B. Note that, all hysteresis operators have the value of 1 in the region S+, and the value of 0 in the region S0.
Thus, equation (1) can be adapted to beq(t)=∫∫S+μ(α,β)γαβ[u(t)]dαdβ  (2)
The weight function μ(α,β) is usually determined empirically by a series of increasing and then decreasing inputs to the actuator. It can be identified by differentiating equation (2) twice, and then evaluate the measured data.
Concerning an arbitrary input history shown in FIG. 6A and its corresponding diagram in the α-β plane shown in FIG. 6B, the displacement of an actuator can be expressed below.q(t)=∫∫S1μ(α,β)γαβ[u(t)]dαdβ+∫∫S2μ(α,β)γαβ[u(t)]dαdβ+∫∫S3μ(α,β)γαβ[u(t)]dαdβΔ[X(α1,β0)−X(α1,β1)]+[X(α2,β1)−X(α2,β2)]+X(u(t),β2)  (3)where X(.) is called the Preisach function. Equation (3) can be rearranged to have
                              q          ⁡                      (            t            )                          =                                            ∑                              k                =                1                                            n                -                1                                      ⁢                          [                                                X                  ⁡                                      (                                                                  α                        k                                            ,                                              β                                                  k                          -                          1                                                                                      )                                                  -                                  X                  ⁡                                      (                                                                  α                        k                                            ,                                              β                        k                                                              )                                                              ]                                +                      X            ⁡                          (                                                u                  ⁡                                      (                    t                    )                                                  ,                                  β                                      n                    -                    1                                                              )                                                          (        4        )            
For example, X(α,β)=qα−qαβ, where qα is the measured displacement after the input voltage is increased from 0 to α and qαβ is the measured displacement after the input voltage is decreased from α to β.
Another case for the input voltage u(t) on the descending curve shown in FIG. 7, the equation (4) can be expressed as
                              q          ⁡                      (            t            )                          =                                            ∑                              k                =                1                                            n                -                1                                      ⁢                          [                                                X                  ⁡                                      (                                                                  α                        k                                            ,                                              β                                                  k                          -                          1                                                                                      )                                                  -                                  X                  ⁡                                      (                                                                  α                        k                                            ,                                              β                        k                                                              )                                                              ]                                +                      [                                          X                ⁡                                  (                                                            α                      n                                        ,                                          β                                              n                        -                        1                                                                              )                                            -                              N                ⁡                                  (                                                            α                      n                                        ,                                          u                      ⁡                                              (                        t                        )                                                                              )                                                      ]                                              (        5        )            
To implement the Preisach model, the classical hysteresis effect profile 20a (Preisach functions) are measured experimentally from the piezoelectric actuators for different combinations of (α,β), which shown in FIG. 8. For points of the (α,β) plane lying within any of the squares or triangles, the bilinear spline interpolation can be used to solve the problems.
For example, the displacement of points a, b, and c can be represented as
(1) Point a: the displacement after the input voltage is increased from 0V to 60Vqa=X(60,0)(2) Point b: the displacement after the input voltage is increased from 0V to 105V, and then decreased from 105V to 90Vqb=X(105,0)−X(105,90)(3) Point c: the displacement after the input voltage is increased from 0V to 135V, decreased from 135V to 45V, and then increased from 45V to 120Vqc=[X(135,0)−X(135,45)]+X(120,45)
Please note, FIG. 8 as an example:
α0=0, α1=15, α2=30, . . . α10=150
β0=0, β1=15, β2=30, . . . β10=150
Because X(α,β) means
“the displacement after the input voltage is increased from 0 to α”                Minus        
“the displacement after the input voltage is decreased from α to β”
Therefore, α>β.
So, the value of X(α,β) only exists on the upper triangle area, which means the classical hysteresis effect profile 20a only memorizes the meshed nodal points.
In addition, when α=β, X(α,β)=0
Please also note that when β=0, X(α,0) means “the displacement after the input voltage is increased from 0 to α”
Although the classical Preisach model claims that the experiment must be under the condition of very slow motion; however, in fact, the displacement output would not be zero, which is shown in FIG. 9. The establishment of hysteresis model by use of classical Preisach model may be inaccurate. In other words, in the classical Preisach model, the classical hysteresis effect profile 20a only memorizes “the displacement after the input voltage is increased from 0 to α”, and does not memorize “the displacement after the input voltage is increased from 0 to α, and then decreased from α to 0.
Therefore, when the experiment is not under the condition of very slow motion of the actuator, the classical Preisach model will be failed.
Furthermore, under the different speed of actuator's motion, the hysteresis effect profile should be different.